MATHEMATICS

Provenmath - mathematical project which states the foundations of mathematics and derives mathematical knowledge from these foundations. Every theorem on Provenmath is proved within the pages of Provenmath and can be traced down to the foundations. The content of Provenmath is self-contained. If you have found a theorem which interests you then you can be certain that you can also find a complete proof of this theorem.

Geometry - This section is devoted to geometry. It contains proofs of some facts known from school. We are not concerned with axioms here. Instead we try to solve problems using simple facts that can be accepted by everyone who can deal with paper, pencil, line and circle.

Goedel's Incompleteness Theorem - On this page we give an outline of the proof of Goedel's Incompleteness Theorem. We construct a statement in set theory such that it is not a theorem and its negation is not a theorem. What we show on this page is not a presentation of the proof that was written by Goedel himself. We present all the basic ideas that are necessary to understand what the theorem asserts and what it would take to write a proof.

Divisibility by Three Explained - On this page we prove the theorem known from school that an integer is divisible by 3 if and only if the sum of its digits is divisible by 3. We intend our proof to be understandable for everyone who has basic familiarity with integer numbers and who is capable of concentrating his attention.

Heine continuity implies Cauchy continuity without the Axiom of Choice - On this page we state and prove that every Heine continuous real function is also Cauchy continuous. In our proof we do not use the Axiom of Choice.

The set of all real numbers is uncountable - On this page we prove that it is false that there exists a 1-1 and "onto" function from |N to |R.

Proof that metric spaces are paracompact - one page PDF file and the LaTeX source code.

Math ASCII Notation - mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts. The purpose of this page is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity...

Continuous Nowhere Differentiable Function
Proof of the existence of a continuous function f:[0,1]->[0,1] that is nowhere differentiable (uses the Banach Contraction Principle).

Continuous Nowhere Monotonic Function
Proof of the existence of a continuous function f:[0,1]->|R that is neither increasing nor decreasing on any subinterval of [0,1] (uses the Baire Category Theorem).

Topological Space - this page contains the definitions of the following concepts: topological space, open set, closed set, interior, closure. Contains the most basic theorems about these concepts together with an easy-to-read proof which is slightly informal in some places.

Michal Ryszard Wojcik's PhD Thesis - Closed and connected graphs of functions; examples of connected punctiform spaces, Katowice 2008

Transformer Language Model Mathematical Definition - A clean no-jargon mathematical definition of transforrmer language model with a Python implementation that focuses on clarity rather than efficiency.